## N 80n 70n 60n 50n 40n 30n 20n 10n

Fig. 5.55 a-d Transport of the western boundary current driven by a point sink specified at location 0q, in units of the total sink Sq.

Fig. 5.55 a-d Transport of the western boundary current driven by a point sink specified at location 0q, in units of the total sink Sq.

interior of each basin water uniformly moves upward, as assumed in the model, and this model-assumed uniform upwelling drives poleward flow, as dictated by the linear potential vorticity balance.

### 5.2.3 Generalized theories of deep circulation

Since Stommel and his colleagues proposed a theoretical framework for the deep circulation in the world's oceans in the 1960s, their theory has dominated the field of deep circulation in the world's oceans. It was only in the 1980s that people started to realize the limitations of the classical theory of deep circulation.

Stommel's theory is based on a very simple and solid theoretical foundation, including the steadiness of the circulation, the assumption of no bottom topography, and the assumption of uniform upwelling and point sources of deep water. Under such assumptions, the simple solution obtained from the model is the only logical result from the fundamentals of fluid dynamics.

It was very clear that direct observational confirmation of the poleward flow in the deep ocean interior predicted by the theory was very difficult because it is extremely slow. Thus, for a long time the relatively fast western boundary currents observed in the world's oceans have been used as the most concrete evidence in support of the theory. However, it eventually became clear that the simple assumptions made in the theory have rather serious limitations if one wants to describe the deep circulation in the world's oceans more accurately.

Modification of the Stommel andArons theory With the progress of in situ observations and further scrutiny of the physics involved, the discrepancy between the classical Stommel theory and the circulation in the deep ocean became clearer, and more realistic features have been added to the model in order to describe different physical features of the deep circulation. The most essential issues include the following.

Eastern boundary currents due to topographic ft-effect

The bottom slope along the eastern boundary of the basin may be so steep that the corresponding topographic j-effect overpowers the planetary j-effect. As a result, a strong deep boundary current can appear along the eastern boundary of the basin.

### Non-uniform upwelling

Although upwelling is the term used in many studies, strictly speaking "diapycnal velocity" is the correct term. In general, upwelling indicates a positive vertical velocity, which can be induced by diapycnal mixing; it can also be induced by adiabatic vertical motions due to flow over topography. In addition, strong upwelling can also be associated with the divergence of Ekman transport, such as the coastal upwelling and the strong upwelling in the Southern Ocean. In the following discussion, upward motion due to diapycnal mixing is referred to as upwelling.

For the idealized basin with a flat bottom, upwelling velocity is not necessarily uniform in the whole basin. Simple scaling suggests that the vertical velocity is related to the ther-mocline depth w = K/h, where k is the vertical diffusivity and h is the scale depth. Since the thermocline is shallow along the eastern boundary, it is expected that upwelling there can be much larger than in the western part of the basin.

Kawase (1987) postulated that the upwelling through the interface of a two-layer model is linearly proportional to the deviation from a prescribed reference position of the interface w* = r [ho(x,y) — h(t,x,y)] (5.86)

where r is the relaxation factor, which is the inverse of the relaxation time, and h0(x,y) and h(t, x, y) are the depth of the reference state and the current state of the interface. As an alternative, the interface upwelling rate can be determined as a part of the solution through coupling the upwelling with the surface thermohaline forcing (Huang, 1993a).

Recent field observations indicate that diapycnal mixing is greatly enhanced near rough bottom topography, such as the mid-ocean ridge and seamounts (Ledwell et al, 2000). Thus, the corresponding upwelling specified in the inverse reduced-gravity model should be non-uniform. Indeed, the dynamical consequence of non-uniform mixing/upwelling is a research frontier in abyssal circulation; a topic which is discussed in detail later in this chapter.

Baroclinic circulation in a stratified abyssal ocean

The classical Stommel and Arons theory is based on the assumption that density is uniform in the abyss, and that it predicts the barotropic circulation in the abyssal ocean. In a stratified ocean, density stratification gives rise to baroclinic circulation. The meridional velocity is governed by the Sverdrup relation, i.e., the linear vorticity balance jv = f dw/dz. In a zonal section, density along the eastern boundary is relatively light, and this density anomaly propagates westward by stationary diffusive Rossby waves; thus, baroclinic structure in velocity and density exists in the abyss (Pedlosky, 1992).

### Hypsometry

In general, the bottom of an individual basin is not flat; instead of a flat bottom, most basins can be characterized in terms of the shape of a Chinese wok. Due to the decrease in horizontal area with depth, the circulation can be quite different from the classical theory of Stommel.

Assume that the horizontal area of the basin is A(z) and the total amount of deepwater source for the abyss below level z is S (z). Thus, the vertical velocity across level z is w = S (z) /A (z). For the interior ocean, the linear potential vorticity balance is jv = fw = fddz (Ajly). Different combinations of S(z) and A(z) can give rise to dramatically different meridional velocity patterns at different levels (Rhines and McCready, 1989). For example, if dw/dz < 0, meridional velocity in the abyssal interior must move toward lower latitudes, opposite to the direction predicted by the classical theory.

### Geothermal heating

Although the thermal isolation condition has been used in most ocean models, there is a heat flux through the sea floor. Geothermal heat flux released from major eruptions of volcanic activity may induce large plumes over the mid-ocean ridge, as discussed by Stommel (1982). However, for the global thermohaline circulation, the contribution from geothermal heat flux is small, and thus negligible. On the other hand, if we are interested in the abyssal circulation, geothermal heat flux can be a major contributor to the abyssal stratification, as demonstrated by Thompson and Johnson (1996). Adcroft et al. (2001) carried out numerical experiments using an oceanic general circulation model. By including a uniformly distributed geothermal heat flux of 50 mW/m2, the bottom temperature increases by 0.1 - 0.3°C, compared with the case without geothermal heat flux. Such a change in bottom water temperature is substantial; thus, it is clear that if we want to simulate bottom water properties and circulation accurately, geothermal heat flux must be included.

### Grounding

Since the value of bottom water flux is finite, part of the bottom of the deep basin may not be covered by bottom water. The finiteness of bottom water makes the problem quite different from the cases discussed in the Stommel theory (Speer and McCartney, 1992). If the strength of the bottom water source is not great enough, say

a phenomenon known as "grounding" occurs (w* is the specified upwelling velocity through the upper surface of the bottom water); an example is shown in Figure 5.57.

This phenomenon is a mirror image of the outcropping phenomenon discussed in Section 4.1.4. The essential difference between the models for outcropping and for grounding is in the integral constraint. For the generalized Parsons model, the total amount of water in the upper layer must be constant, while in the grounding model, the total amount of upwelling should equal the source of bottom water.

a A-A Section view a A-A Section view

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